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Numerical Schemes Obtained From Lattice Boltzmann Equations For Advection Diffusion Equations

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  • SHINSUKE SUGA

    (Social and Environmental Systems Division, National Institute for Environmental Studies, 16-2 Onogawa, Tsukuba, Ibaraki, 305-8506, Japan)

Abstract

Stability and accuracy of the numerical schemes obtained from the lattice Boltzmann equation (LBE) used for numerical solutions of two-dimensional advection-diffusion equations are presented. Three kinds of velocity models are used to determine the moving velocity of particles on a squre lattice. A system of explicit finite difference equations are derived from the LBE based on the Bhatnagar, Gross and Krook (BGK) model for individual velocity model. In order to approximate the advecting velocity field, a linear equilibrium distribution function is used for each of the moving directions. The stability regions of the schemes in the special case of the relaxation parameter ω in the LBE being set toω=1are found by analytically solving the eigenvalue problems of the amplification matrices corresponding to each scheme. As for the cases of general relaxation parameters, the eigenvalue problems are solved numerically. A benchmark problem is solved in order to investigate the relationship between the accuracy of the numerical schemes and the order of the Peclet number. The numerical experiments result in indicating that for the scheme based on a 9-velocity model we can find the parameters depending on the order of the given Peclet number, which generate accurate solutions in the stability region.

Suggested Citation

  • Shinsuke Suga, 2006. "Numerical Schemes Obtained From Lattice Boltzmann Equations For Advection Diffusion Equations," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 17(11), pages 1563-1577.
    • Handle: RePEc:wsi:ijmpcx:v:17:y:2006:i:11:n:s0129183106010030
    DOI: 10.1142/S0129183106010030
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    Cited by:

    1. Oleg Ilyin, 2023. "Hybrid Lattice Boltzmann Model for Nonlinear Diffusion and Image Denoising," Mathematics, MDPI, vol. 11(22), pages 1-12, November.
    2. Ilyin, Oleg, 2023. "Lattice Boltzmann model for diffusion equation with reduced truncation errors: Applications to Gaussian filtering and image processing," Applied Mathematics and Computation, Elsevier, vol. 456(C).

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